Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for beginner pupils in their first years of college or even in high school.
Nevertheless, grasping how to handle these equations is essential because it is primary information that will help them navigate higher arithmetics and advanced problems across various industries.
This article will discuss everything you need to master simplifying expressions. We’ll review the principles of simplifying expressions and then validate our skills via some practice problems.
How Does Simplifying Expressions Work?
Before you can learn how to simplify expressions, you must grasp what expressions are to begin with.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can contain numbers, variables, or both and can be connected through subtraction or addition.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions containing coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is essential because it lays the groundwork for understanding how to solve them. Expressions can be expressed in complicated ways, and without simplification, everyone will have a hard time trying to solve them, with more opportunity for a mistake.
Undoubtedly, every expression be different concerning how they're simplified depending on what terms they include, but there are common steps that can be applied to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Simplify equations within the parentheses first by adding or using subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term outside with the one on the inside.
Exponents. Where feasible, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, use multiplication and division to simplify like terms that apply.
Addition and subtraction. Finally, use addition or subtraction the remaining terms in the equation.
Rewrite. Make sure that there are no additional like terms that require simplification, then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few additional properties you should be informed of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and leave the variables as [[is|they are]-70. For example, the expression 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and keeping the x as it is.
Parentheses that include another expression outside of them need to utilize the distributive property. The distributive property gives you the ability to to simplify terms outside of parentheses by distributing them to the terms on the inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is called the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution rule applies, and each individual term will will require multiplication by the other terms, resulting in each set of equations, common factors of one another. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses denotes that the negative expression will also need to be distributed, changing the signs of the terms inside the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign outside the parentheses will mean that it will be distributed to the terms on the inside. Despite that, this means that you can eliminate the parentheses and write the expression as is due to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous principles were easy enough to implement as they only dealt with properties that impact simple terms with numbers and variables. However, there are additional rules that you need to apply when working with exponents and expressions.
In this section, we will discuss the properties of exponents. Eight properties influence how we utilize exponents, that includes the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 is equal to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent won't change in value. Or a1 = a.
Product Rule. When two terms with the same variables are apply multiplication, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient will subtract their two respective exponents. This is expressed in the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equal to the inverse of that term over 1. This is expressed with the formula a-m = 1/am; (a/b)-m = (b/a)m.
Power of a Power Rule. If an exponent is applied to a term already with an exponent, the term will end up having a product of the two exponents that were applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that have different variables needs to be applied to the respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will take the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that says that any term multiplied by an expression on the inside of a parentheses should be multiplied by all of the expressions on the inside. Let’s watch the distributive property used below.
Let’s simplify the equation 2(3x + 5).
The distributive property states that a(b + c) = ab + ac. Thus, the equation becomes:
2(3x + 5) = 2(3x) + 2(5)
The expression then becomes 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have multiple rules that you have to follow.
When an expression consist of fractions, here's what to keep in mind.
Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions separately by their denominators and numerators.
Laws of exponents. This shows us that fractions will typically be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest should be included in the expression. Apply the PEMDAS principle and ensure that no two terms share matching variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to the expressions inside the parentheses, while PEMDAS will govern the order of simplification.
As a result of the distributive property, the term on the outside of the parentheses will be multiplied by each term on the inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, be sure to add the terms with the same variables, and each term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation like this:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule states that the you should begin with expressions inside parentheses, and in this scenario, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed to the two terms inside the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for the moment and simplify the terms with factors associated with them. Remember we know from PEMDAS that fractions require multiplication of their denominators and numerators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple because any number divided by 1 is that same number or x/1 = x. Thus,
y(5x)/4
5xy/4
The expression y/4(2) then becomes:
y/4 * 2/1
2y/4
Thus, the overall expression is:
1/3x + 5xy/4 + 2y/4
Its final simplified version is:
1/3x + 5/4xy + 1/2y
Example 3
Simplify the expression: (4x2 + 3y)(6x + 1)
In exponential expressions, multiplication of algebraic expressions will be utilized to distribute all terms to each other, which gives us the equation:
4x2(6x + 1) + 3y(6x + 1)
4x2(6x) + 4x2(1) + 3y(6x) + 3y(1)
For the first expression, the power of a power rule is applied, which tells us that we’ll have to add the exponents of two exponential expressions with the same variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no remaining like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey the distributive property, PEMDAS, and the exponential rule rules and the rule of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Solving and simplifying expressions are very different, but, they can be part of the same process the same process due to the fact that you have to simplify expressions before solving them.
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